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Reflectance Magnitude

We have shown that the conical cap reflectance has no poles in the strict right-half plane. For passivity, we also need to show that its magnitude is bounded by unity for all $ s$ on the $ j\omega $ axis.

We have

$\displaystyle R_J(j\omega) = \frac{1 - e^{-2j\omega} - 2j\omega e^{-2j\omega}}{... ...mega} - 1 + 2j\omega} = e^{-2j\omega} \frac{\overline{D(j\omega)}}{D(j\omega)} $

so that $ \left\vert R_J(j\omega)\right\vert = 1$, which is exactly lossless, as expected.

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Poles at
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Stability Proof